Algebra ( Period 5)

Chapter 3-1 Graphing Linear Equations

There are FOUR types of linear graphs and this chapter begins with an  OVERALL, BIG  picture

Positive Slope- slants up from left to right
Negative Slope- slants down from left to right
Horizontal line- stays flat from left to right ( constant function)
Vertical Line- stays straight up and down ( Not a function—why??)

Somethings to look for:
Domain
Range
End behavior
Intercepts
Extrema
Positive/Negative
Increasing/Decreasing
Symmetry

A Linear Equation is an equation that forms a line when it is graphed. Linear equations are often written in the form Ax + By = C
This is called standard form. In this equation C is called  a constant  Ax and By are variable terms.

A ≥ 0
A and B BOTH cannot be 0
A, B,and C are ALL integers with a GCF= 1

If you see a term such as xy attached to together it cannot be a linear equation. If the exponent on a variable is different than the understood 1,  it is not a linear equation

in 3x + 2y = 5

A = 3
B = 2
C = 5

In x = -7 ( Yes that is in Standard Form)

A = 1
B = 0
C = -7

Identify Linear Equations

Determine whether each equation is  a linear equation. Write the equation in Standard form
y = 4 – 3x   YES

To put this equation in standard form, we need to move the -3x term to the other side, using the  Addition Property of Equality and the Additive Inverse Property.  So that the x and y values are on the SAME side and the constant is always on the other side to the right of the equal sign.
3x + y = 4
A = 3
B = 1
C = 4

6x –xy = 4   NO

the term xy has two variables the equation cannot be written in AX + By = C . It is not a linear equation

(1/3)y = -1  Yes

It becomes y = -3

A= 0
B = 1
C = -3

A linear equation can be represented on a coordinate graph. The x- coordinate of the point at which the graph of the equation crosses the x-axis is called the x-intercept. The y- coordinate of the point at which the graph of the equation crosses the y-axis is called the y-intercept.

The graph of  linear equation has AT MOST one x- intercept and ONE y-intercept ( unless it is the equation x = 0, which is the y-axis or y = 0, which is the x-axis. In those two special cases every number is a y-intercept or an x-intercept, respectively)

Real World Example  Swimming Pool Page 157 in your textbook

A swimming pool is being drained at a rate of 720 gallons per hour. The table on Page 157 shows the function relating the volume of water in a pool and the time in hours that the pool has been draining.

Find the x- and y- intercepts on the graph of the function.

Looking at the table we see that the x intercept is 14 ( that is when y is 0)
and the y-intercept is 10,080 ( that is the value of y, when x = 0)

Describe what the intercepts mean in this situation: This should remind you of our unit at the beginning of the year!

The x intercept 14 means that after 14 hours the pool is completed drained because it has a volume of 0 gallons!

The y- intercept of 10,080 means that the pool contained 10,080 gallons of water at time 0 ( or before it started to drain)

Graph by Using Intercepts
Graph 2x + 4y = 16 using just the x-intercept and y-intercept

2x + 4(0) = 16   replace y with 0 (or as taught in class cover over the y value and solve)

2x = 16 so x = 8 ( when y = 0) ( 8,0)

This means the graph intersects the x-axis at (8,0)

Now

2(0) + 4y = 16  replace x with 0 ( or as taught in class- cover over the x value and solve)

4y = 16
y = 4  ( when x = 0)  ( 0, 4)

This means the graph intersect the x-axis at (0, 4)

Plot these two point and draw a line through them

Notice that this has both an x- intercept and  y-intercept

Some lines have only an x- intercept and NO y-intercept  or vice versa

y = b is a horizontal line that has only a y- intercept (unless b=0)

The graph of x = a is a vertical line that has only an x- intercept (unless a = 0)

Lines that are neither vertical or horizontal cannot have more than one x- and/or y-intercept.

Graphing Using an XY Table

Another way to graph is choosing random x values , plugging those into the equation to find the corresponding y values, and graphing those points you found.

Although 2 points determine a line, it is always best to find 3 points so that you are sure you did not make a mistake on either of the first two points.

If the coefficient of x is a fraction, select a value that is  multiple of the denominator so hopefully you won’t end up with fractions to graph!